Question

Is it me or whoever answered my question makes no sense?

http://www.peeranswer.com/study/mathematics/557e2b603102d10905d35bd5

My question:

Answer 1

There is this property that states that \[\int_k^kf(x)~\mathrm dx = 0\quad \forall~ k\in\mathbb R\]

From what I understand, integral is area enclosed by \(f(x)\) and x-axis. So this property makes sense since \(0\cdot f(k)=0\), given that width of line is \(0\).

So saying we have \(f(x)=\dfrac{1}{x}\), is \(\displaystyle \int_0^0 f(x)~\mathrm dx \) still \(0\)?

Because \(f(x)\) doesn't exist at \(x=0\) hence integral doesn't exist since it is basically \(0\cdot\text{indetermine}\)

So I believe property is supposed to be \[\int_k^kf(x)~\mathrm dx = \begin{cases}0 & \text{if }~f(k)~\exists\\\not\exists & \text{if }~f(k)~\not\exists \end{cases}\]

Right? If not, why not?

Answer 2

@freckles @SithsAndGiggles